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D₃×#547 #86699

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

Splittings

Fitting	C₃×#547 ⋊ C₂
Sylow	C₃ ⋊ C₂×#547

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₃
commutator, socle, cosocle	C₂²×C₃	C₂⁵
Fitting	C₃×#547	C₂
hypercenter	#547	D₃
hypocenter	C₃	C₂×#547
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ #547

1632 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 5 generators on 15 points:

(1,2,3)(10,15)(13,14);(5,7)(8,11,12,9)(10,14,15,13);(8,10)(9,13)(11,14)(12,15);(4,5)(6,7)(8,9,12,11)(10,14,15,13);(2,3)

Conjugacy classes & character table

75 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#1625, D₃×#1635, D₃×#1647, …
maximal normal subgroup of	D₃×#1625, D₃×#1635, D₃×#1647, …
maximal quotient of	D₃×#2339, D₃×#2344, D₃×D₄×C₄⋊C₄, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 19275);